From the first relation, we get OD + DC = p. From the next two, after substituting the values of OD and DC, we get xcosα + ysinα = p

We’re done !

The required equation is xcosα + ysinα = p. (That was quick.)

Remember what we did (and what we’ll always do) when deriving the equation. Take any point P(x,y) on the line and establish a relation between x and y (and the given constants), which will always hold true.

Another method..

Method II

This time, I’ll use the intercept form of the line (this one) to derive the normal form of the equation. Have a look at the figure below.

Now, what I’ve done is, expressed the intercepts in terms of the given information (p, α).

Since we now know the intercepts, we can use the intercept form of the equation, i.e. \(\frac{x}{psecα}+\frac{y}{pcosecα}=1\)

And, we get the same equation back, which is xcosα + ysinα = p

Lesson Summary

The equation of the line, whose perpendicular distance from origin is p, and this perpendicular makes an angle α with the X-axis, is given by xcosα + ysinα = p.